Question: Let $t$ be a parameter that varies over all real numbers.  Any parabola of the form
\[y = 3x^2 + tx - 2t\]passes through a fixed point.  Find this fixed point.
Answer: To obtain the fixed point, we want to eliminate $t$ in the equation
\[y = 3x^2 + tx - 2t.\]We can do so by taking $x = 2.$  This leaves us with $y = 3 \cdot 2^2 = 12,$ so the fixed point is $\boxed{(2,12)}.$